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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000326
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 01 => 2
([],2)
=> [1,1]
=> 110 => 011 => 2
([(0,1)],2)
=> [2]
=> 100 => 001 => 3
([],3)
=> [1,1,1]
=> 1110 => 0111 => 2
([(1,2)],3)
=> [2,1]
=> 1010 => 0101 => 2
([(0,2),(1,2)],3)
=> [2,2]
=> 1100 => 0011 => 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 0001 => 4
([],4)
=> [1,1,1,1]
=> 11110 => 01111 => 2
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01101 => 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> 11010 => 01011 => 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 11100 => 00111 => 3
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 0011 => 3
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 11100 => 00111 => 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 10100 => 00101 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 111100 => 001111 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => 00011 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 00001 => 5
([],5)
=> [1,1,1,1,1]
=> 111110 => 011111 => 2
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 011101 => 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 110110 => 011011 => 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 111010 => 010111 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 111100 => 001111 => 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 01011 => 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 111010 => 010111 => 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 11100 => 00111 => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 011001 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 111100 => 001111 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 101010 => 010101 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 001101 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 1111010 => 0101111 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 1111100 => 0011111 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 110010 => 010011 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 001101 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 110100 => 001011 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 11111100 => 00111111 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 111000 => 000111 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 111100 => 001111 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 00101 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 001101 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => 00011 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 1111100 => 0011111 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 1011100 => 0011101 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 111000 => 000111 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 110100 => 001011 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 100100 => 001001 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 101000 => 000101 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 1101100 => 0011011 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 1111000 => 0001111 => 4
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St001322
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([],2)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([],3)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 4 - 1
([],4)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4 = 5 - 1
([],5)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ? = 3 - 1
([(0,1),(0,3),(1,2),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(1,2),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ? = 3 - 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St000297
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 10 => 1 = 2 - 1
([],2)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,1]
=> 110 => 2 = 3 - 1
([],3)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1010 => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 1100 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [4]
=> 10000 => 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 10100 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 11000 => 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 1100 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 11000 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> 11100 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 100100 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 101000 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 10100 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 101000 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> 11000 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> 1010000 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 1100000 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,2,2]
=> 101100 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 110100 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> 11000000 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 11100 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 1100000 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 110100 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [2,2,2,1]
=> 111010 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [4,4,2]
=> 1100100 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [4,4,4]
=> 1110000 => 3 = 4 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> [8,6]
=> 1001000000 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> 1010000000 => ? = 2 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,1]
=> [9,8]
=> 10100000000 => ? = 2 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> 1010000000 => ? = 2 - 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,1]
=> [7,6,1]
=> 1010000010 => ? = 2 - 1
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> 1010000000 => ? = 2 - 1
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> 1010000000 => ? = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,1]
=> [9,8]
=> 10100000000 => ? = 2 - 1
([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(1,4),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,4),(0,6),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,1]
=> [7,6,6]
=> 1011000000 => ? = 2 - 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,5),(1,3),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,1),(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,1),(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,1]
=> [7,6,1]
=> 1010000010 => ? = 2 - 1
([(0,6),(1,4),(1,5),(2,3),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,4),(0,5),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,3),(0,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,3,3,3]
=> [6,6,6,1]
=> 1110000010 => ? = 4 - 1
([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,3,3,1]
=> [7,6,6]
=> 1011000000 => ? = 2 - 1
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,4,4,4,2]
=> [5,5,4,4]
=> ? => ? = 3 - 1
([(0,4),(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> 1110000000 => ? = 4 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 86%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 2 - 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 3 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? = 4 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> []
=> ? = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [4,3]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [5,5]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? = 6 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [4,3]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2 = 3 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 4 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> ? = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? = 7 - 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 2 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> ? = 2 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 2 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,2,1]
=> [8,7]
=> ? = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 2 - 1
([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 4 - 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,1]
=> [3,3,3,3,3,1]
=> [6,5,5]
=> ? = 2 - 1
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 4 - 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 4 - 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 2 - 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 4 - 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 2 - 1
([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 3 - 1
([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> ? = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 4 = 5 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 3 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [6,6,2]
=> [[1,2,5,6,7,8],[3,4,11,12,13,14],[9,10]]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [5,4,4]
=> [[1,2,3,4,13],[5,6,7,8],[9,10,11,12]]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [6,6,2]
=> [[1,2,5,6,7,8],[3,4,11,12,13,14],[9,10]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2,2]
=> [6,6,3]
=> [[1,2,3,7,8,9],[4,5,6,13,14,15],[10,11,12]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [6,6,4]
=> [[1,2,3,4,9,10],[5,6,7,8,15,16],[11,12,13,14]]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 5 - 1
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 4 - 1
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,3,3,3]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,3,3]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [4,4,3,3]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 5 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8]]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 4 = 5 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13]]
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 2 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 3 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12],[13]]
=> ? = 3 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [[1,2,11],[3,4,14],[5,6],[7,8],[9,10],[12,13]]
=> [[1,3,5,7,9,12],[2,4,6,8,10,13],[11,14]]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> [[1,2,5,8,11],[3,6,9,12],[4,7,10,13]]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [[1,2,11],[3,4,14],[5,6],[7,8],[9,10],[12,13]]
=> [[1,3,5,7,9,12],[2,4,6,8,10,13],[11,14]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6,15],[7,8],[10,11],[13,14]]
=> [[1,3,5,7,10,13],[2,4,6,8,11,14],[9,12,15]]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9,16],[11,12],[14,15]]
=> [[1,3,5,8,11,14],[2,4,6,9,12,15],[7,10,13,16]]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> [[1,4,7,10,13,16],[2,5,8,11,14,17],[3,6,9,12,15,18]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? = 5 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 4 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12],[13]]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 3 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001184
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> [3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001481
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> [3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St001803
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 0 = 2 - 2
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 0 = 2 - 2
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 2 - 2
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 2 - 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 3 - 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 3 - 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 2 - 2
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 2 - 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0 = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 3 - 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 2 - 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0 = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 2 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 4 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 4 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 2 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 3 = 5 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 6 - 2
([],6)
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 2 - 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 2 - 2
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> 0 = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 2 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 2 - 2
([(2,5),(3,4),(4,5)],6)
=> [2,2,2,1,1]
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> 0 = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,1,1]
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> ? = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 2 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> ? = 2 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,1]
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> ? = 2 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 4 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 2 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,1]
=> [5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> ? = 2 - 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 2 - 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,1]
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> ? = 2 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 3 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> ? = 3 - 2
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001316The domatic number of a graph. St001621The number of atoms of a lattice. St001330The hat guessing number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition.
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